Irreducible Representations « Abstract Nonsense

The Irreps of the Product of Finitely Many Finite Groups

Point of post: In this post we shall discuss how one can find the set of all irreps, up to equivalence, of a group of the form $G_1\times\cdots\times G_n$ given the irreps of $G_k$ for $k\in[n]$ .

Motivation

We’ve developed quite an extensive theory regarding how to find the irreps of a finite group $G$ and how to relate those irreps to one another, as well as their characters.The question remains though if there is a natural way that the irreps of $G$ relate naturally to constructions based on $G$ . In particular, in this post we are interested in determining the relationships between the irreps (in particular the characters) of the product $G\times H$ of two groups $G$ and $H$ given the knowledge of the characters of $G$ and $H$ . So for example, it’s easy (as we’ve shown) to construct the character table for $S_3$ and it’s equally easy to construct the character table for $\mathbb{Z}_2$ . A next logical step would be to combine them and find the character table for $S_3\times\mathbb{Z}_2$ . It would be nice if one would have to not go tromping through all the extra work to create this (larger) character table having gone through the (admittedly small amount of work) to construct the ones for $S_3$ and $\mathbb{Z}_2$ . In this post we shall show that our greatest wish is true–we can’t just easily get some characters of $S_3\times\mathbb{Z}_2$ from those of $S_3$ and $\mathbb{Z}_2$ but we can easily get all of them.

Character Table of S_3 By Finding the Irreducible Representations

Point of post: In this post we construct the first of a few character tables, namely we construct the character table for $S_3$ .

Motivation

We now start off nice and easy and construct the classic character table for $S_3$ using the techniques from the last post. $S_3$ may be perhaps the easiest charater table to construct, but it will give us a good start to stretch our proverbial legs. In this post though, we find the character table using minimal machinery by actually constructing the irreducible characters of $S_3$ instead of using the techniques in our previous post. This method is, in my opinion, for the purpose of character table construction, not preferable. Indeed, one must actually come up with representatives from each equivalency class of irreps of $S_3$ . This post shall be useful to illustrate how beautifully simple the construction of character tables is made by the theory we’ve developed.

March 22, 2011 Posted by Alex Youcis | Algebra, Group Theory, Representation Theory | Character Table, Character Table For S3, Character Table For S_3, Character Theory, Irreducible Representations, Representation Theory | 2 Comments

Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. II)

Point of post: This post is a continuation of this one.

January 18, 2011 Posted by Alex Youcis | Algebra, Representation Theory | Direct Sum of Representations, Invariant Subspaces, Irreducible Representations, Irreps, Representation Theory, Subrepresentations | 6 Comments

About me

My name is Alex Youcis. I am currently a senior at the University of Maryland, College park getting my bachelor’s in mathematics. Before I came to UMD I was a math major at Drexel University in Philadelphia, Pennsylvania. I was born and raised in Lititz, Pennsylvania, a small town near Lancaster.

As is probably clear, I really enjoy doing math. It’s a passion for me, and I don’t means this in the weekend-hobby sense, I can’t imagine doing anything else other than mathematics. That said, I’m not one of those space cadets one often sees wandering around math departments, trying to solve others problems, or perpetrating some other equally creepy shenanigans. No, while math is a gargantuan aspect of my life, there are many, many other things that interest me. Perhaps the most consuming of these is my love of music. I find it difficult to go even one day without my iPhone, or grooveshark.com If you want to say something to me in the car, you have about eight seconds before my iPhone is plugged in and we’ve set sail on a musical adventure. I have hundreds, upon hundreds of bands that I intensely love, but at the time of writing I can emphatically say that Regina Spektor is my favorite artist–the woman is a musical Gauss. I also love dogs, friends, conversations about things, craft beer and the occasional online comic.

If you’d like to know anything about this blog see the pertinent page in all its splendorous glory.

If you have any questions regarding myself, math, music, or anything else feel free to e-mail me at alex.youcis@gmail.com

Current Course List:

Advanced Microeconomics

Numerical Analysis–uses Suli and Meyers

Riemann Surfaces–Uses Varolin. This is available, in near entirety, for free here.

Algebraic Number Theory–Using various course notes, including those of Milne. The course website is here.

Representation Theory of $p$ -adic Groups–This shall use various notes, the most intimidating of which is this manuscript of Bernstein, the least intimidating of which are these notes of Ban.

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Abstract Nonsense

Crushing one theorem at a time

The Irreps of the Product of Finitely Many Finite Groups

Character Table of S_3 By Finding the Irreducible Representations

Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. II)

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